Confluent Heun functions and the Coulomb problem for spin 1/2 particle in Minkowski space

TL;DR

This paper explores solutions to the quantum Coulomb problem for a spin 1/2 particle using confluent Heun functions, offering alternative methods to the traditional hypergeometric function approach and confirming a consistent energy spectrum.

Contribution

It introduces new solution methods for the Coulomb problem using confluent Heun functions and establishes relations between hypergeometric and Heun function solutions.

Findings

Multiple solution methods using confluent Heun functions are viable.

All methods yield the same energy spectrum, confirming their validity.

Relations between hypergeometric and confluent Heun functions are established.

Abstract

In the paper, the well-known quantum mechanical problem of a spin 1/2 particle in external Coulomb potential, reduced to a system of two first-order differential equations, is studied from the point of view of possible applications of the Heun function theory to treat this system. It is shown that in addition to the standard way to solve the problem in terms of the confluent hypergeometric functions (proposed in 1928 by G. Darvin and W. Gordon), there are possible several other possibilities which rely on applying the confluent Heun functions. Namely, in the paper there are elaborated two combined possibilities to construct solutions: the first applies when one equation of the pair of relevant functions is expressed trough hypergeometric functions, and another constructed in terms of confluent Heun functions. In this respect, certain relations between the two classes of functions are established. It is shown that both functions of the system may be expressed in terms of confluent Heun functions. All the ways to study this problem lead us to a single energy spectrum, which indicates their correctness.